175 research outputs found

### The dimension of the range of a transient random walk

We find formulas for the macroscopic Minkowski and Hausdorff dimensions of
the range of an arbitrary transient walk in Z^d. This endeavor solves a problem
of Barlow and Taylor (1991).Comment: 37 pages, 5 figure

### Strong invariance and noise-comparison principles for some parabolic stochastic PDEs

We consider a system of interacting diffusions on the integer lattice.
By letting the mesh size go to zero and by using a suitable scaling, we
show that the system converges (in a strong sense) to a solution of the
stochastic heat equation on the real line. As a consequence, we obtain
comparison inequalities for product moments of the stochastic heat equa-
tion with different nonlinearities

### Dynamical percolation on general trees

H\"aggstr\"om, Peres, and Steif (1997) have introduced a dynamical version of
percolation on a graph $G$. When $G$ is a tree they derived a necessary and
sufficient condition for percolation to exist at some time $t$. In the case
that $G$ is a spherically symmetric tree, H\"aggstr\"om, Peres, and Steif
(1997) derived a necessary and sufficient condition for percolation to exist at
some time $t$ in a given target set $D$. The main result of the present paper
is a necessary and sufficient condition for the existence of percolation, at
some time $t\in D$, in the case that the underlying tree is not necessary
spherically symmetric. This answers a question of Yuval Peres (personal
communication). We present also a formula for the Hausdorff dimension of the
set of exceptional times of percolation.Comment: 24 pages; to appear in Probability Theory and Related Field

### A macroscopic multifractal analysis of parabolic stochastic PDEs

It is generally argued that the solution to a stochastic PDE with
multiplicative noise---such as $\dot{u}=\frac12 u"+u\xi$, where $\xi$ denotes
space-time white noise---routinely produces exceptionally-large peaks that are
"macroscopically multifractal." See, for example, Gibbon and Doering (2005),
Gibbon and Titi (2005), and Zimmermann et al (2000). A few years ago, we proved
that the spatial peaks of the solution to the mentioned stochastic PDE indeed
form a random multifractal in the macroscopic sense of Barlow and Taylor (1989;
1992). The main result of the present paper is a proof of a rigorous
formulation of the assertion that the spatio-temporal peaks of the solution
form infinitely-many different multifractals on infinitely-many different
scales, which we sometimes refer to as "stretch factors." A simpler, though
still complex, such structure is shown to also exist for the
constant-coefficient version of the said stochastic PDE.Comment: 41 page

### Fractal-dimensional properties of subordinators

This work looks at the box-counting dimension of sets related to subordinators (non-decreasing Lévy processes). It was recently shown in Savov (Electron Commun Probab 19:1–10, 2014) that almost surely limδ→0U(δ)N(t,δ)=t , where N(t,δ) is the minimal number of boxes of size at most δ needed to cover a subordinator’s range up to time t, and U(δ) is the subordinator’s renewal function. Our main result is a central limit theorem (CLT) for N(t,δ) , complementing and refining work in Savov (2014). Box-counting dimension is defined in terms of N(t,δ) , but for subordinators we prove that it can also be defined using a new process obtained by shortening the original subordinator’s jumps of size greater than δ . This new process can be manipulated with remarkable ease in comparison with N(t,δ) , and allows better understanding of the box-counting dimension of a subordinator’s range in terms of its Lévy measure, improving upon Savov (2014, Corollary 1). Further, we shall prove corresponding CLT and almost sure convergence results for the new process

### Patterns in random walks and Brownian motion

We ask if it is possible to find some particular continuous paths of unit
length in linear Brownian motion. Beginning with a discrete version of the
problem, we derive the asymptotics of the expected waiting time for several
interesting patterns. These suggest corresponding results on the
existence/non-existence of continuous paths embedded in Brownian motion. With
further effort we are able to prove some of these existence and non-existence
results by various stochastic analysis arguments. A list of open problems is
presented.Comment: 31 pages, 4 figures. This paper is published at
http://link.springer.com/chapter/10.1007/978-3-319-18585-9_

### Breadth first search coding of multitype forests with application to Lamperti representation

We obtain a bijection between some set of multidimensional sequences and this
of $d$-type plane forests which is based on the breadth first search algorithm.
This coding sequence is related to the sequence of population sizes indexed by
the generations, through a Lamperti type transformation. The same
transformation in then obtained in continuous time for multitype branching
processes with discrete values. We show that any such process can be obtained
from a $d^2$ dimensional compound Poisson process time changed by some integral
functional. Our proof bears on the discretisation of branching forests with
edge lengths

### Composition of processes and related partial differential equations

In this paper different types of compositions involving independent
fractional Brownian motions B^j_{H_j}(t), t>0, j=1,$ are examined. The partial
differential equations governing the distributions of
I_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|), t>0 and
J_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|^{1/H_1}), t>0 are derived by different methods
and compared with those existing in the literature and with those related to
B^1(|B^2_{H_2}(t)|), t>0. The process of iterated Brownian motion I^n_F(t), t>0
is examined in detail and its moments are calculated. Furthermore for
J^{n-1}_F(t)=B^1_{H}(|B^2_H(...|B^n_H(t)|^{1/H}...)|^{1/H}), t>0 the following
factorization is proved J^{n-1}_F(t)=\prod_{j=1}^{n} B^j_{\frac{H}{n}}(t), t>0.
A series of compositions involving Cauchy processes and fractional Brownian
motions are also studied and the corresponding non-homogeneous wave equations
are derived.Comment: 32 page

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